Eurostar says it is hoping to put on two-thirds of its services - for 26,000 passengers - today.

However, only those who had tickets for Eurostar on Saturday or Sunday will be eligible for travel.

The first train left Paris at 7.30am and another left St Pancras at 8am.

Trains will run between 7.30am and 6.30pm and the firm is advising passengers that if their journeys are not essential they should rebook for a later date or have their tickets refunded.

It said in a statement: "Customers holding tickets for Monday, December 21, and Tuesday, December 22, will be eligible for travel on Wednesday, December 23. Customers holding tickets for Wednesday 23 or Thursday 24 December will be eligible for travel on Thursday, December 24.

All services were cancelled on Monday thanks to a forecast of more snow in France and tests on the trains.

Eurotunnel's Folkestone passenger check-in was closed from lunchtime due to the massive number of people trying to get trains who had previously been booked on Eurostar.

It remains closed as of 6.30pm and all day-trip bookings are still being turned away. Eurotunnel says the situation will be reassessed once the backlog of traffic has been cleared.

Anyone booked on on the service should check with its website before setting out. A normal freight service is being provided.

SeaFrance is carrying cars and passengers transferred from Eurotunnel and the firm has also offered to help stranded Eurostar passengers and has made extra space available. LD Lines' Dover - Boulogne service will additionally be carrying foot passengers on certain advertised sailings to help.

Eurostar says that as a result of the test runs conducted on Sunday it believes it has established the cause of the failure of its trains. It says every year it carries out a 'winterisation' programme of the entire fleet which until now has worked successfully.

But in a statement it added: "We now know, however, that we need to further enhance the snow screens and snow shields in the power cars of the trains."

The problems came with trains that had travelled through very cold weather in France. As they entered the relatively warm Channel Tunnel, condensation formed and ice melted, dripping into electrical systems and causing the failures.

The company apologised also announced it had commissioned an independent review of the problems.

The review, which will start immediately, will be led by Christopher Garnett - former chief executive of GNER and formerly commercial director of Eurotunnel - in conjunction with a French inspector.

A company statement added: "We strongly advise passengers whose travel is non-essential to change their booking to a later date or claim a refund on their tickets. For customers who have been unable to travel over the last few days, Eurostar will cover the cost of out-of pocket expenses, for example, hotels and taxis.